Continuum robots and control thereof

ABSTRACT

Method for controlling continuum robots and systems therefrom are provided. In the system and method, a new system of equations is provided for controlling a shape of the elastic member and a tension on a tendon applying a force to an elastic member of the robot. The system of equations can be used to estimate a resulting shape of the elastic member from the tension applied to the tendon. The system of equations can also be used to estimate a necessary tension for the tendon to achieve a target shape.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of Provisional application Ser. No.61/362,353 entitled “CONTINUUM ROBOTS AND CONTROL THEREOF”, filed Jul.8, 2010, which is herein incorporated by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates to continuum robots, and more specificallyto apparatus and methods for.

BACKGROUND

Continuum robots offer a number of potential advantages over traditionalrigid link robots in certain applications, particularly those involvingreaching through complex trajectories in cluttered environments or wherethe robot must compliantly contact the environment along its length. Theinherent flexibility of continuum robots makes them gentle to theenvironment, able to achieve whole arm manipulation, and gives rise to aunique form of dexterity—the shape of the robot is a product of bothactuator and externally applied forces and moments. Thus, kinematicmodels which consider the effects of external loading have been activeareas of recent research, and models that consider pneumatic actuation,multiple flexible push-pull rods, and a elastic member consisting ofconcentric, pre-curved tubes, have recently been derived.

Cosserat rod theory has shown promise as a general tool for describingcontinuum robots under load, but application of the theory totendon-actuated continuum robots has not yet been fully explored.Simplified beam mechanics models have been widely used to successfullyobtain free-space kinematic models for tendon-actuated robots. Theconsensus result is that when the tendons are tensioned, the elasticmember assumes a piecewise constant curvature shape. This approach isanalytically simple and has been thoroughly experimentally vetted onseveral different robots. However, this approach is limited in that itcannot be used to predict the large spatial deformation of the robotwhen subjected to additional external loads. Cosserat rod theoryprovides the modeling framework necessary to solve this problem, andinitial work towards applying it to tendon actuated robots has beenperformed by considering planar deformations and using the simplifyingassumption that the load from each tendon consists of a single pointmoment applied to the rod at the termination arc length. However, suchmodels are limited.

SUMMARY

Embodiments of the invention concern systems and method for controllingcontinuum robots. In a first embodiment of the invention, a continuumrobot is provided. The continuum robot includes an elastic member, aplurality of guide portions disposed along the length of the elasticmember, and at least one tendon extending through the plurality of guideportions. In the continuum robot, the tendon is arranged to extendthrough the plurality of guide portions to define a tendon path, wherethe tendon is configured to apply a deformation force to the elasticmember via the plurality of guide portions, and where the tendon pathand an longitudinal axis of the elastic member are not parallel.

In a second embodiment of the invention, a method for managing acontinuum robot is provided. In the method, the robot includes anelastic member, a plurality of guide portions disposed along the lengthof the elastic member, and at least one tendon extending through theplurality of guide portions. In the continuum robot, the tendon isarranged to extend through the plurality of guide portions to define atendon path, where the tendon is configured to apply a deformation forceto the elastic member via the plurality of guide portions. The methodincludes the steps of applying a tension to the at least one tendon andcomputing the resulting shape of the elastic member resulting from saidtension by solving a system of equations. In the method, the system ofequations is given by

$\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix}\overset{.}{v} \\\overset{.}{u}\end{bmatrix}}} = {{\begin{bmatrix}{D + A} & G \\B & {C + H}\end{bmatrix}^{- 1}\begin{bmatrix}d \\c\end{bmatrix}}.}}$

where u is the deformed curvature vector consisting of the angular ratesof change of the attached rotation matrix R with respect to arc lengths, v is a vector comprising linear rates of change of the attached framewith respect to arc length s, C and D are stiffness matrices for theelastic member, and matrices A, B, G, H, are functions of the tensionapplied to the at least one tendon, the tendon path, and itsderivatives, d is vector based on the external force on the elasticmember, and c is vector based on an external moment on the elasticmember.

In a third embodiment of the invention, a method for managing acontinuum robot is provided. In the method, the robot includes anelastic member, a plurality of guide portions disposed along the lengthof the elastic member, and at least one tendon extending through theplurality of guide portions. In the continuum robot, the tendon isarranged to extend through the plurality of guide portions to define atendon path, where the tendon is configured to apply a deformation forceto the elastic member via the plurality of guide portions. The methodincludes the steps of determining a target shape for the elastic memberand computing a tension for the at least one tendon to provide thetarget shape by evaluating a system of equations. In the method, thesystem of equations is given by

$\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix}\overset{.}{v} \\\overset{.}{u}\end{bmatrix}}} = {{\begin{bmatrix}{D + A} & G \\B & {C + H}\end{bmatrix}^{- 1}\begin{bmatrix}d \\c\end{bmatrix}}.}}$

where u is the defotmed curvature vector consisting of the angular ratesof change of the attached rotation matrix R with respect to arc lengths, v is a vector comprising linear rates of change of the attached framewith respect to arc length s, C and D are stiffness matrices for theelastic member, and matrices A, B, G, H, are functions of the tensionapplied to the at least one tendon, the tendon path, and itsderivatives, d is vector based on the external force on the elasticmember, and c is vector based on an external moment on the elasticmember.

In a fourth embodiment of the invention, a continuum robot is provided.The robot includes an elastic member, a plurality of guide portionsdisposed along the length of the elastic member, and at least one tendonextending through the plurality of guide portions and arranged to extendthrough the plurality of guide portions to define a tendon path, whereinthe at least one tendon is configured to apply a deformation force tothe elastic member via the plurality of guide portions. The robot alsoincludes an actuator for applying a tension to the at least one tendon;and a processing element for using a system of equations for controllinga shape of the elastic member and the tension. The system of equationsis given by:

$\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix}\overset{.}{v} \\\overset{.}{u}\end{bmatrix}}} = {{\begin{bmatrix}{D + A} & G \\B & {C + H}\end{bmatrix}^{- 1}\begin{bmatrix}d \\c\end{bmatrix}}.}}$

where u is the deformed curvature vector consisting of the angular ratesof change of the attached rotation matrix R with respect to arc lengths, v is a vector comprising linear rates of change of the attached framewith respect to arc length s, C and D are stiffness matrices for theelastic member, and matrices A, B, G, H, are functions of the tensionapplied to the at least one tendon, the tendon path, and itsderivatives, d is vector based on the external force on the elasticmember, and c is vector based on an external moment on the elasticmember.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of the results of simulations of a continuumrobot with single, straight, tensioned tendons with in-plane andout-of-plane forces applied at the tip.

FIG. 2A is an illustration of an exemplary of robot shape/workspacemodification for a robot with four straight tendons spaced at equalangles around its periphery.

FIG. 2B is an illustration of an exemplary of robot shape/workspacemodification for a robot with four helical tendons that each make onefull revolution around the shaft.

FIG. 3 is an illustration of an arbitrary section of rod from c to ssubject to distributed forces and moments, showing the internal forces nand moments m.

FIG. 4 is an illustration of a general cross section of the continuumrobot material or support disk, showing tendon locations.

FIG. 5 is an illustration of a small section of a rod showing how theforce distribution that the tendon applies to its surrounding medium isstatically equivalent to a combination of force and moment distributionson the elastic member itself.

FIG. 6 is an illustration schematically showing (a) the coupled Cosseratrod and tendon approach that includes all of the tendon loads and (b)the point moment approach only includes the attachment moment.

FIG. 7 is an x-y-z plot of the time response of a Continuum robot with ahelical tendon is simulated for a step input in tendon tension.

FIG. 8 is shows an exemplary continuum robot configured to operate inaccordance with the various embodiments, where the inset shows adetailed view of the tendon guide portions.

FIG. 9 is an x-y-z plot of the actual and simulated result plane loadingof an exemplary continuum robot using a straight tendon.

FIG. 10 is an x-y-z plot of the actual and simulated result ofout-of-plane loading of an exemplary continuum robot using a straighttendon.

FIG. 11 is an x-y-z plot of the actual and simulated result ofout-of-plane loading for an exemplary continuum robot using a straighttendon and high tension.

FIG. 12A is an x-y-z plot of the actual and simulated results ofoperation of an exemplary continuum robot using a helical tendon withouta load.

FIG. 12B is an x-y-z plot of the actual and simulated results ofoperation of an exemplary continuum robot using a helical tendon with atip load.

FIG. 13 is an x-y-z plot of the actual and simulated results ofoperation of an exemplary continuum robot using a polynomial tendon withtip loads according to Table II.

FIG. 14 is a schematic illustration of a continuum robot system 1400 inaccordance with the various embodiments.

FIG. 15 shows an exemplary system 1500 includes a general-purposecomputing device 1500 for performing methods and processes in accordancewith the various embodiments.

DETAILED DESCRFPTION

The present invention is described with reference to the attachedfigures, wherein like reference numerals are used throughout the figuresto designate similar or equivalent elements. The figures are not drawnto scale and they are provided merely to illustrate the instantinvention. Several aspects of the invention are described below withreference to example applications for illustration. It should beunderstood that numerous specific details, relationships, and methodsare set forth to provide a full understanding of the invention. Onehaving ordinary skill in the relevant art, however, will readilyrecognize that the invention can be practiced without one or more of thespecific details or with other methods. In other instances, well-knownstructures or operations are not shown in detail to avoid obscuring theinvention. The present invention is not limited by the illustratedordering of acts or events, as some acts may occur in different ordersand/or concurrently with other acts or events. Furthermore, not allillustrated acts or events are required to implement a methodology inaccordance with the present invention.

I. Introduction

The various embodiments of the invention provide systems and methods forthe control of tendon-actuated continuum robots. In particular, thevarious embodiments of the invention extend previous work on theCosserat rod-based approach by taking into account not only theattachment point moment, but also the attachment point force and thedistributed wrench that the tendon applies along the length of theelastic member. This approach couples the classical Cosserat string androd models to express tendon loads in terms of the rod's kinematicvariables.

The difference between this new coupled model and the point moment modelfor out of plane loads is shown in FIG. 1, and provide an experimentalcomparison of the two models described below in Sec. V. FIG. 1 is anillustration of the results of simulations of a continuum robot withsingle, straight, tensioned tendons with in-plane and out-of-planeforces applied at the tip. These illustrate the difference between themodel proposed in this paper which includes distributed tendon wrenches,and the commonly used point moment approximation. For planardeformations and loads, the two models differ only by axial compression(which is small in most cases). However, for out of plane loads, theresults differ significantly and including distributed wrenches enhancesmodel accuracy (see Sec. V).

The various embodiments thus provide two new innovations overconventional methods. First, a new Cosserat rod-based model is providedfor the spatial deformation of tendon actuated continuum robots undergeneral external point and distributed wrench loads. This model is thefirst to treat the full effects of all of the tendon loads in ageometrically exact way for large 3D deflections. Second, the new modelis the first to describe the mechanics of general tendon routing pathsthat need not run straight (along the undeformed robot configuration),as has been the case in prior prototypes. Thus, by providing a generalmodel that can address most, if not all, types of tendon routing, thisexpands the design space and the set of shapes achievable fortendon-actuated robots.

In view of the foregoing, the various embodiments provide systems andmethods for controlling continuum robots using exact models for theforward kinematics, statics, and dynamics and with general tendonrouting experiencing external point and distributed loads, The modelsaccount for lance deformations due to bending, torsion, shear, andelongation. The static model is formulated as a set of nonlineardifferential equations in standard form, and the dynamic model consistsof a system of hyperbolic partial differential equations.

Using this approach, one can accurately predict the shape of a physicalprototype with both straight and non-straight tendon routing paths andwith external loading. With calibrated parameters, the mean tip errorwith respect to the total robot length can be significantly reduced ascompared to conventional methods.

As illustrated in FIGS. 2A and 2B, the design space of achievable robotshapes can be expanded by considering alternative tendon paths. Forexample, FIG. 2A is an illustration of an exemplary of robotshape/workspace modification for a robot with four straight tendonsspaced at equal angles around its periphery. In contrast, FIG. 2B is anillustration of an exemplary of robot shape/workspace modification for arobot with four helical tendons that each make one full revolutionaround the shaft, in accordance with an enibodiment of the invention.The two designs differ significantly in tip orientation capability, andthe helical design may be better suited to some types of tasks, e.g. aplanar industrial pick and place tasks or surgical tasks. Thus, suchcontinuum robots could be used to enhance the capabilities of medicalinstruments introduced using an orifice or minimal incision and allowadditional control during procedures. Some types of devices can includedevices for proceudes in the throat and airways (introduced via themouth), in the colon (introduced via the anus), in the stomach(introduced via the mouth and traveling through the esophogus), in theabdomen (either via a transgastric natural orifice transluminalendoscopic surgery approach, or via an incision in the abdomen similarto normal laparoscopic surgery), in skull base surgery (entering via thenose), in the brain and subarachnoid (entering an area around the brainstem and center of the spine) spaces (entering via a craniotomy), or inthe bladder and kidneys (entering via the urethra).

The models in accordance with the various embodiments of the inventiontherefore allow new quasi-static and/or dynamic control techniques fortendon-actuated continuum robots in the future. Furthermore, theinclusion of general external loads in tendon actuated continuum robotmodels is an important step forward for future practical applications,given their significant sag under self-weight and when carryingpayloads. Additionally, such models can be used to address the issue ofmodeling static friction, and real-time computation of static anddynamic robot shape.

II. Model for a Simple Cosserat Rod

A. Rod Kinematics

In Cosserat-rod theory, a rod is characterized by its centerline curvein space p(s)∈

³ and its material orientation, R(s)∈SO(3) as functions of a referenceparameter s∈[0 L]. Thus a homogeneous transformation can be used todescribe the entire rod:

${g(s)} = \begin{bmatrix}{R(s)} & {p(s)} \\0 & 1\end{bmatrix}$

Kinematic variables v(s) and u(s) represent the linear and angular ratesof change of g(s) with respect to s expressed in coordinates of the“body frame” g(s). Thus, the evolution of g(s) along s is defined by thefollowing relationships;

{dot over (R)}(s)=R(s)û(s), {dot over (p)}(s)=R(s)v(s)  (1)

where, the dot denotes a derivative with respect to s, and the ̂ and{hacek over ( )} and operators are as defined by R. M. Murray, Z. Li,and S. S. Sastry in “A Mathematical Introduction to RoboticManipulation.” Boca Raton, Fla.: CRC Press, 1994. See also theDerivation Appendix for an explanation of these operators.

Letting the undeformed reference configuration of the rod be g*(s),where the z axis of R*(s) is chosen to be tangent to the curve p*(s).One could use the Frenet-Serret or Bishop's convention to define the xand y axes of R*(s), or, if the rod has a cross section which is notradially symmetric, it is convenient to make the x and y axes align withthe principal axes. The reference kinematic variables v* and u* can thenbe obtained by [v*^(T) u*^(T)]^(T)=(g*⁻¹(s)ġ*

If the reference configuration happens to be a straight cylindrical rodwith s as the arc length along it, then v*=[0 0 1]^(T) and u*(s)=[0 00]^(T).

B. Equilibrium Equations

One can the write the equations of static equilibrium for an arbitrarysection of rod as shown in FIG. 3. The internal force and moment vectors(in global frame coordinates) are denoted byn and the applied forcedistribution per unit of s is f, and the applied moment distribution perunit of s is l. Taking the derivative of the static equilibriumconditions with respect to s, one arrives at the classic forms of theequilibrium differential equations for a special Cosserat rod,

{hacek over (n)}(s)+f(s)=0,  (2)

{dot over (m)}(s)+{dot over (p)}(s)×n(s)+l(s)=0.  (3)

C. Constitutive Laws

The difference between the kinematic variables in the rod's referencestate and those in the deformed state can be directly related to variousmechanical strains. For instance, transverse shear strains in thebody-frame x and y directions correspond to v_(x)-v_(x)* andv_(y)-v_(y)* respectively, while axial elongation or stretch in thebody-frame z direction corresponds to v_(z)-v_(z)*. Similarly, bendingstrains about the local x and y axes are related to u_(x)-u_(x)* andu_(y)-u_(y)* respectively, while torsional strain about the local z axisis related to u_(z)-u_(z)*.

One can use linear constitutive laws to map these strain variables tothe internal forces and moments. Assuming that the x and y axes of g*are aligned with the principal axes of the cross section, one obtains

n(s)=R(s)D(s)(v(s)−v*(s)),

m(s)=R(s)C(s)(u(s)−u*(s)),  (4)

where

D(s)=diag (GA(s), GA(s), EA(s)), and

C(s)=diag (EI _(xx)(s), EI _(yy)(s), EI_(xx)(s)+EI _(yy)(s)),

where A(s) is the area of the cross section, E(s) is Young's modulus,G(s) is the shear modulus, and I_(xx)(s) and I_(yy)(s) are the secondmoments of area of the tube cross section about the principal axes.(Note that I_(xx)(s)+I_(yy)(s) is the polar moment of inertia about thecentroid.) One can use these linear relationships here because they arenotationally convenient and accurate for many continuum robots, but theCosserat rod approach does not require it.

D. Explicit Model Equations

Equations (2) and (3) can then be written in terms of the kinematicvariables using equation (4), their derivatives, and equation (1). Thisleads to the full set of differential equations shown below.

{dot over (p)}=Rv

{dot over (R)}=Rû

{dot over (v)}={dot over (v)}*−D ⁻¹((ûD+{dot over (D)})(v−v*)+R ^(T) f)

{dot over (u)}={dot over (u)}*−C ⁻¹((ûC+Ċ)(u−u*)+{circumflex over(v)}D(v−v*)+R ^(T) l)  (5)

Alternatively, an equivalent system can be obtained using in and n asstate variables rather than v and u.

{dot over (p)}=R(D ⁻¹ R ^(T) n+v*)

{dot over (R)}−R(C ⁻¹ R ^(T) m+u*)⁻

{dot over (n)}=−f

{dot over (m)}=−{dot over (p)}×n−l  (6)

Boundary conditions for a rod which is clamped at s=0 and subject to anapplied force F_(l) and moment L_(l) at s=l would be R(0)=R₀, p(0)=p₀,m(l)=L_(l), and n(l)=F_(l)

III. Coupled Cosserat Rod & Tendon Model

Having reviewed the classic Cosserat-rod model, the derivation anewmodel for tendon driven continuum manipulators in accordance with thevarious embodiments of the invention will now be presented. Thederivation uses the Cosserat model of Section II to describe the elasticmember and the classic Cosserat model for extensible strings to describethe tendons. For purposes of the model, the string and rod models arecoupled together by deriving the distributed loads that the tendonsapply to the elastic member in terms of the rod's kinematic variables,and then incorporating these loads into the rod model.

A. Assumptions

Two standard assumptions are employee in the derivation. First, anassumption of frictionless interaction between the tendons and thechannel through which they travel. This implies that the tension isconstant along the length of the tendon. Frictional forces are expectedto increase as the curvature of the robot increases due to larger normalforces, but the assumption of zero friction is valid if low frictionmaterials are used, which is the case for the experimental prototypediscussed below. Second, the locations of the tendons within the crosssection of the robot are assumed not to change during the deformation.This assumption is valid for designs which use embedded sleeves orchannels with tight tolerances, as well as designs which use closelyspaced tendon guide portions.

B. Tendon Kinematics

One can separate the terms f and l in the equations in (5) into trulyexternal distributed loads, f_(e) and l_(e), and distributed loads dueto tendon tension, f_(t) and l_(t).

f=f _(e) +f _(t)

l=l _(e) +l _(t).  (7)

In order to derive f_(t) and l_(t), one starts by defining the path inwhich the tendon is routed along the robot length. Note that this pathcan be defined by channels or tUbes within a homogeneous elasticatructure, or support disks on an elastic member—both of which affordconsiderable flexibility in choosing tendon routing. In the experimentalprototype, many holes are drilled around the periphery of each supportdisk, allowing easy reconfiguration of tendon path as desired.

A convenient way to mathematically describe the tendon routing path isto define the tendon location within the robot cross section as afunction of the reference parameter s. Thus, the routing path of thei^(th) tendon is defined by two functions x_(i)(s) and y_(i)(s) thatgive the body-frame coordinates of the tendon as it crosses the x-yplane of the attached elastic member frame at s. As shown in FIG. 4, avector from the origin of the attached frame to the tendon location isthen given in attached frame coordinates by

r _(i)(s)=[x _(i)(s)y _(i)(s)0]^(T).  (8)

The parametric space curve defining the tendon path in the global framewhen the robot is in its undeformed reference configuration is thengiven by

p _(i)*(s)=R*(s)r _(i)(s)+p*(s).

Similarly, when the robot is deformed due to tendon tension or externalloads, the new tendon space curve will be

p _(i)(s)=R(s)r _(i)(s)+p(s).  (9)

C. Distributed Forces on Tendons

The governing differential equations for an extensible string can bederived by taking the derivative of the static equilibrium conditionsfor a finite section. This results in the same equation for the internalforce derivative as in equation (2).

{dot over (n)} _(i)(s)+f _(i)(s)=0.  (10)

where f_(i)(s) is the distributed force applied to the i^(th) tendon perunit of s, and n_(i)(s) is the internal force in the tendon. In contrastto a Cosserat rod, an ideal string has the defining constitutiveproperty of being perfectly flexible, meaning it cannot support internalmoments or shear forces, but only tension which is denoted by τ_(i).This requires that the internal force be always tangent to the curvep_(i)(s). Thus, one can write

$\begin{matrix}{{n_{i}(s)} = {\tau_{i}{\frac{{\overset{.}{p}}_{i}(s)}{{{\overset{.}{p}}_{i}(s)}}.}}} & (11)\end{matrix}$

If friction were present, τ_(i) would vary with s, but under thefrictionless assumption, it is constant along the length of the tendon.Using (10) and (11) one can derive the following expression for thedistributed force on the tendon (see Appendix for Derivation):

$\begin{matrix}{{f_{i}(s)} = {{- {\overset{.}{n}}_{i}} = {\tau_{i}\frac{{\overset{\hat{.}}{p}}_{i}^{2}}{{{\overset{.}{p}}_{i}}^{3}}{{\overset{¨}{p}}_{i}.}}}} & (12)\end{matrix}$

D. Tendon Loads on Elastic Member

One can now write the collective distributed loads f_(t) and l_(t) thatthe tendons apply to the elastic member, in terms of the individualforces on the tendons and their locations in. the elastic membercross-section. The total distributed force is equal and opposite to thesum of the individual force distributions on the tendons shown inequation (12), namely,

$f_{t} = {- {\sum\limits_{i = 1}^{n}{f_{i}.}}}$

The distributed moment at the elastic member centroid is the sum of thecross products of each moment arm with each force. Thus,

$l_{t} = {{- {\sum\limits_{i = 1}^{n}{\left( {p_{i} - p} \right)^{\hat{}}f_{i}}}} = {- {\sum\limits_{i = 1}^{n}{\left( {Rr}_{i} \right)^{\hat{}}{f_{i}.}}}}}$

Substituting equation (12), yields

$\begin{matrix}{{f_{t} = {- {\sum\limits_{i = 1}^{n}{\tau_{i\;}\frac{{\overset{\hat{.}}{p}}_{i}^{2}}{{{\overset{.}{p}}_{i}}^{3}}{\overset{¨}{p}}_{i}}}}},{l_{t} = {- {\sum\limits_{i = 1}^{n}{{\tau_{i}\left( {Rr}_{i} \right)}^{\hat{}}\frac{{\overset{\hat{.}}{p}}_{i}^{2}}{{{\overset{.}{p}}_{i}}^{3}}{{\overset{¨}{p}}_{i}.}}}}}} & (13)\end{matrix}$

One can then express these total force and moment distributions in termsof the kinematic variables u, v, R and p so that one can substitute theminto equations (7) and (5). To do this, one expands {dot over (p)} and{umlaut over (p)}. Differentiating equation (9) twice yields,

{dot over (p)} _(i) =R(ûr _(i) +{dot over (r)} _(i) +v),

{umlaut over (p)} _(i) =R(û(ûr _(i) +{dot over (r)} _(i) +v)+{dot over(û)}r _(i) +û{dot over (r)} _(i) +{umlaut over (r)} _(i) +{dot over(v)}).  (14)

It is noted that {umlaut over (p)} is a function of {dot over (u)} and{dot over (v)}. Therefore, substituting these results into equation(13), and equation (13) into the rod model equation (5) via equation(7), one can obtain an implicitly defined set of differential equations.Fortunately, the resulting equations are linear in u and v, and it istherefore possible to manipulate them into an explicit form. Rewritingthem in this way (such that they are amenable to standard numnicalmethods) is the topic of the following subsection.

E. Explicit Decoupled Model Equations

The coupled rod & tendon model is given in implicit form by equations(5), (7), (13), and (14). In this subsection, these implicit equationsare manipulated into explicit, order, state-vector form. To express theresult concisely, some intermediate matrix and vector quantities aredefined, starting with equation (14) expressed in body-framecoordinates, i.e.

{dot over (p)} _(i) ^(b) =ûr _(i) +{dot over (r)} _(i) +v.

{umlaut over (p)} _(i) ^(b) =û{dot over (p)} _(i) ^(b) +{dot over (û)}r_(i) +û{dot over (r)} _(i) +{umlaut over (r)} _(i) +{dot over (v)}.

Now define Matrices A_(i), A, B_(i), and B, as well as vectors a_(i), a,b_(i), and b, as follows:

$\begin{matrix}{{A_{i} = {{- \tau_{i}}\frac{\left( {\overset{\hat{.}}{p}}_{i}^{b} \right)^{2}}{{{\overset{.}{p}}_{i}^{b}}^{3}}}},} & {{B_{i} = {{\hat{r}}_{i}A_{i}}},} \\{{A = {\sum\limits_{i = 1}^{n}A_{i}}},} & {{B = {\sum\limits_{i = 1}^{n}B_{i}}},} \\{{a_{i} = {A_{i}\left( \; {{\hat{u}{\overset{.}{p}}_{i}^{b}} + {\hat{u}{\overset{.}{r}}_{i}} + {\hat{r}}_{i}} \right)}},} & {{b_{i} = {{\hat{r}}_{i}a_{i}}},} \\{{a = {\sum\limits_{i = 1}^{n}a_{i}}},} & {{b = {\sum\limits_{i = 1}^{n}b_{i}}},}\end{matrix}$

to find that f_(t) and l_(t) can now be expressed as

$\begin{matrix}{{f_{t} = {R\left( {a + {A\overset{.}{v}} + {\sum\limits_{i = 1}^{n}{A_{i}\overset{\hat{.}}{u}r_{i}}}} \right)}},{l_{t} = {{R\left( {b + {B\overset{.}{v}} + {\sum\limits_{i = 1}^{n}{B_{i}\overset{\hat{.}}{u}r_{i}}}} \right)}.}}} & (15)\end{matrix}$

The vector terms Σ_(i=1) ^(n) A_(i){dot over (û)}r_(i) and Σ_(i=1)^(n)B_(i){dot over (û)}r_(i) are both linear in the elements of {dotover (u)}. Therefore, it is possible to express them both by equivalentlinear operations on {dot over (u)}. That is, one can define matrices Gand H as

$G = {\sum\limits_{i = 1}^{n}\begin{bmatrix}{A_{i}{\hat{e}}_{1}r_{i}} & {A_{i}{\hat{e}}_{2}r_{i}} & {A_{i}{\hat{e}}_{3}r_{i}}\end{bmatrix}}$ $H = {\sum\limits_{i = 1}^{n}\begin{bmatrix}{B_{i}{\hat{e}}_{1}r_{i}} & {B_{i}{\hat{e}}_{2}r_{i}} & {B_{i}{\hat{e}}_{3}r_{i}}\end{bmatrix}}$

where e₁, e₂, and e₃ are the standard basis vectors [1 0 0], [0 1 0],and [0 0 1]. Then, equation (15) becomes

f _(t) =R(a+A{dot over (v)}+G{dot over (u)}).

l _(t) =R(b+B{dot over (v)}+H{dot over (u)}).

Substituting tendon load expressions into the last two equations in (5)and rearranging them provides

(D+A){dot over (v)}+G{dot over (u)}=d

B{dot over (v)}+(C+H){dot over (u)}=c

where the vectors c and d are functions of the state variables as shownbelow.

d=D{dot over (v)}*−(ûD+{dot over (D)})(v−v*)−R ^(T) f _(c) −a

c=C{dot over (u)}*−(ûC+Ċ)(u−u*)−{circumflex over (v)}D(v−v*)−R ^(T) l_(c) −b.

One can now easily write the governing equations as

{dot over (p)}=Rv

{dot over (R)}=Rû

$\begin{matrix}{\begin{bmatrix}\overset{.}{v} \\\overset{.}{u}\end{bmatrix} = {{\begin{bmatrix}{D + A} & G \\B & {C + H}\end{bmatrix}^{- 1}\begin{bmatrix}d \\c\end{bmatrix}}.}} & (17)\end{matrix}$

Noting that the quantities on the right hand side of equation (17) aremerely functions of the state variables and system inputs (u, R, τ_(n),f_(e) and l_(e)) one arrives at a system of differential equations instandard explicit form, describing the shape of a continuum robot withany number of generally routed tendons and with general external loadsapplied.

This system can be solved by any standard numerical integration routinefor systems of the form {dot over (y)}=f(s,y). The required matrixinverse may be calculated (either numerically or by obtaining a closedfonn inverse) at every integration step, or one could alternativelyrewrite the equations as a system with a state dependent mass matrix onthe left hand side and use any standard numerical method for solvingM(y,s){dot over (y)}=f(s,y). For purposes of the simulations andexperiments in accordance with the various embodiments of the invention,numerically inversion is used,

F. Boundary Conditions

When tendon i terminates at s=l_(i) along the length of the robot, itapplies a point force to its attachment point equal and opposite to theinternal force in the tendon given by equation (11). Thus, the pointforce vector is given by

$\begin{matrix}{F_{i} = {{- {n_{i}\left( l_{i} \right)}} = {{- \tau_{i}}\frac{{\overset{.}{p}}_{i}\left( l_{i} \right)}{{{\overset{.}{p}}_{i}\left( l_{i} \right)}}}}} & (18)\end{matrix}$

With a moment arm of p_(i)(l_(i))−p(l_(i)), this force creates a pointmoment L_(i) at the elastic member centroid of,

$\begin{matrix}{L_{i} = {{- {\tau_{i}\left( {{R\left( l_{i} \right)}{r_{i}\left( l_{i} \right)}} \right)}^{\hat{}}}{\frac{{\overset{.}{p}}_{i}\left( l_{i} \right)}{{{\overset{.}{p}}_{i}\left( l_{i} \right)}}.}}} & (19)\end{matrix}$

If at some location s=σ, point loads F(σ) and L(σ) (resulting fromtendon terminations or external loads) are applied to the elasticmember, the internal force and moment change across the boundary s=σ by,

n(σ⁻)=n(σ⁺)+F(σ),

m(σ⁻)=m(σ⁺)+L(σ).  (20)

where σ⁻ and σ⁺ denote locations just before and just after s=σ. Anycombination of external point loads and tendon termination loads can beaccommodated in this way.

G. Point Moment Model

In prior tendon robot models, tendon actuation has often been modeled bysimply applying the pure point moment in equation (19) to an elasticmember model at the location where each tendon is attached, withoutconsidering the point force at the attachment point and the distributedtendon loads along the length (see FIG. 6). This approach is convenientbecause it allows one to use the classical Cosserat rod equations bysimply applying boundary conditions that take into account the tendontermination moments.

This approximation for planar robots is justified since the eftcts ofthe point force and the distributed loads effectively “cancel” eachother, leaving only the point moment. Thus, as shown in FIG, I thisapproach yields almost exactly the same final shape as the full coupledmodel when the rohot deformation occurs in a plane.

However, as shown in FIG. 1, the two approaches diverge as the robotshape becomes increasingly non-planar due to a transverse load at thetip. In Section V, an investigation of the accuracy of both approaches aset of experiments on a prototype robot is provided.

IV. Dynamic Model

Based on the coupled rod and tendon model presented above for staticcontinuum robot deformations, a model for the dynamics of a continuumrobot with general tendon routing is derived. Such a model will beuseful for analyzing the characteristics of specific designs as well asthe development of control algorithms similar to those derived forplanar robotswith straight tendons. As shown below, adding the necessarydynamic terms and equations results in a hyperbolic system of partialdifferential equations, which can be expressed in the standard form

y _(t) =f(s,t,y,y _(s)),  (21)

where a subscript s or t is used in this section to denote partialderivatives with respect to the reference parameter s and time trespectively.

Two new vector variables are introduced, q and w, which are the bodyframe linear and angular velocity of the rod at s. These are analogousto u and v respectively, but are defined with respect to time instead ofarc length. Thus,

p _(t) =Rq R _(t) =Rŵ.  (22)

Recalling from equations in (5) that

p _(s) =Rv R _(s) =Rû,  (23)

and using the fact that p_(st)=p_(ts) and R_(st)=R_(ts) one can derivethe following compatibility equations,

u _(t) =w _(s) +ûw v _(t=q) _(s) +ûq−ŵv _(s)  (24)

Equations (2) and (3) describe the static equilibrium of the rod. Todescribe dynamics, one can add the time derivatives of the linear andangular momentum per unit length in place of the zero on the right handside, such that they become,

{dot over (n)}+f=ρAp _(u),  (25)

{dot over (m)}+{dot over (p)}×n+l=δ _(t)(RρJw),  (26)

where ρ is the mass density of the rod, A is the cross sectional area ofthe elastic member, and J is the matrix of second area moments of thecross section, Expanding these and applying the equations in (24) onecan obtain a complete system in the form of equation (21),

$\begin{matrix}{\mspace{79mu} {{{p_{t} = {Rq}}\mspace{20mu} {R_{t} = {R\; \hat{\omega}}}\mspace{20mu} {v_{t} = {q_{s} + {\hat{u}q} - {\hat{\omega}v}}}\mspace{20mu} {u_{t} = {\omega_{s} + {\hat{u}\; \omega}}}{q_{t} = {\frac{1}{\rho \; A}\left( {{D\left( {v_{s} - v_{s}^{*}} \right)} + {\left( {{\hat{u}D} + D_{s}} \right)\left( {v - v^{*}} \right)} + {R^{T}\left( {f_{e} + f_{t}} \right)} - {\rho \; A\hat{\omega}\; q}} \right)}}}{\omega_{t} = {\left( {\rho \; J} \right)^{- 1}\left( {{C\left( {u_{e} - u_{s}^{*}} \right)} + {\left( {{\hat{u}C} + C_{s}} \right)\left( {u - u^{*}} \right)} + {\hat{v}{D\left( {v - v^{*}} \right)}} + {R^{T}\left( {l_{e} + l_{t}} \right)} - {\hat{\omega}\rho \; J\; \omega}} \right)}}}} & (27)\end{matrix}$

where f_(t) and l_(t) can be computed using the equations in (16).Typically, conditions at t=0 are given for all variables along thelength of the robot, and the boundary conditions of Subsection III-Fapply for all times.

A. Dynamic Simulation

To illustrate the capability of the equations in (27) to describe thetime evolution of the shape of a continuum robot with general tendonrouting, the following dynamic simulation of a robot whose elasticmember is identical to that of the experimental prototype described inSection V is provided. The robot contains a single tendon routed in ahelical where the tendon makes one complete revolution around the shaftas it passes from the base to the tip. This routing path is the same asthe one for tendon 5 in the prototype, which is specified in Table 1.

TABLE I TENDON ROUTING PATHS USED IN EXPERIMENTS Tendon (i) 1 2 3 4 5 6x_(i) (mm) 8 0 −8 0 8 cos(2πs/l) refer to (28) y_(i) (mm) 0 8 0 −8 8sin(2πs/l) refer to (28)

FIG. 7 shows snapshots of the robot backbone shape at millisecondintervals after a step input of 5 Newtons of tendon tension was applied.For the numerical simulation Richtmyerts two-step variant of theLax-Wendroff finite-difference scheme was implemented.

The maximum length of the time step for any explicit time-marchingalgorithm for hyperbolic partial differential equations is limited bythe Courant-Friedriechs-Lewy condition for stability. This is a fairlyrestrictive condition for dynamic rod problems because the shear,extension, and torsional vibrations are so fast that a very small isrequired in order to capture them without the simulation becomingunstable. An active research field in mechanics and computer graphicssimulation is to find reduced-order models of rods that are physicallyaccurate and yet capable of being simulated in real-time. Thissimulation confirms the intuition that the elastic member should movetowards a helical very when the helical tendon undergoes a step intension.

V. Experimental Validation

Below are described several different experiments conducted using acontinuum robot prototype with a variety of tendon paths and externalloading conditions applied

A. Prototype Constructions

A prototype robot in accordance with the various embodiments is shown inFIG. 8. The central elastic member 802 is a spring steel rod (ASTM A228)of length l=242 mm and diameter d=0.8 mm with tendon guide portions 804consisting of 12 stand-off disks, 20 mm in diameter, spaced 20 mm apartalong its length. The disks were laser cut from 1.57 um thick PTFEfilled Delrin plastic to minimize friction with the tendons. As shown inthe inset of FIG. 8, 24 small pass-through holes 806 were laser cut in acircular pattern at a radius of 8 mm from the center of each disk. Theelastic member rod 802 was passed through the center holes 808 of thedisks and each was fixed to it using Loctite 401. For tendons 810, 0.36mm PTFE coated fiberglass thread were used. Each tendon 810 was runthrough various pass-through holes along the robot and knotted at theend, after passing through the final support disk. The optimal ratio oftendon support spacing to offset distance from the elastic member wasfound to be 0.4, and the prototype was designed to exactly match thisratio.

Although the exemplary robot configuration utilizes standoff disks toprovide the tendon guide portions, the various embodiments are notlimited in this regard. Rather, any of means of coupling the tendons tothe elastic member to cause deformation of the elastic member can beused in the various embodiments. Further, a particular combination ofmaterials, spacing of guide portions, and openings in the guide portionsis provided, the various embodiments are not limited in this regard.Rather, any variations on the combination recited above can be used withthe various embodiments. Additionally, the methods above can be usedwith any number of tendons. In such embodiments, the tendons can extendalong a same portion of the length of the elastic member or the tendonscan extend over different portions of the length of the elastic member,including overlapping portions.

The tendon routing paths can be reconfigured on this robot by“re-threading” the tendons through a different set of holes in thevarious support disks. The robot's self-weight distribution was measuredto be 0.47 N/m, which is enough to cause significant deformation,producing 44 mm of downward deflection at the tip (18% of the total arclength) for zero tendon tension. This weight was incorporated into allmodel calculations as a distributed force.

B. Experimental Procedure

In each of the following experiments, known tensions were applied totendons behind the base of the robot by passing the tendons overapproximately frictionless pulleys and attaching them to hangingcalibration weights. In those cases with applied point loads, weights812 were also hung from the tip of the robot, as shown in FIG. 8.

In each experiment, a set of 3D elastic member points was collected bymanually touching the elastic member with the tip of an opticallytracked stylus as shown in FIG. 8. A Micron Tracker 2 H3-60 (ClaronTechnology, Inc.) was used to track the stylus, which has a specifiedfiducial measurement accuracy of 0.20 mm.

C. Calibration

The base frame position of the robot can be determined accurately usingthe optically tracked stylus. The angular orientation of the robotelastic member as it leaves the base support plate is more challengingto measure (Note that the elastic member cannot be assumed to exitexactly normal to the plate due to the tolerance between the elasticmember and the hole drilled in the plate, and a 2° angular error in baseframe corresponds to an approximately 8 mm tip error when the robot isstraight). Also, the effective stiffness of the elastic member wasincreased due to the constraints of the standoff disks and Loctiteadhesive at regular intervals. To account for these uncertainties theeffective Young's modulus and the set of XYZ Euler angles (α, B, and γ)describing the orientation of the base frame were calibrated.

The calibration process was accomplished by sorting preconstrainednonlinear optimization problem to find the set of parameters whichminimizes the sum of the positional errors at the tip of the device forthe set of 25 experiments with straight tendon paths described in Sec.V-D and Table II.

TABLE II EXPERIMENTAL TENSIONS AND TIP LOADS Experiments with Tendons1-4 (Straight) Tension (N) 0 0.98 1.96 2.94 2.94 2.94 4.91 Tip Load (N)0 0 0 0 0.098 0.196 0 Experiments with Tendon 5 (Helical) Tension (N)0.98 1.96 2.94 4.91 4.91 4.91 6.87 Tip Load (N) 0 0 0 0 0.098 0.196 0Experiments with Tendon 6 (Polynomial) Tension (N) 1.50 2.46 3.66 4.914.91 Tip Load (N) 0 0 0 0 0.0196

In other words, for the parameter set P={E, α, B, γ}:

$P_{cal} = {\underset{P}{\arg \; \min}\left( {\sum\limits_{k = 1}^{25}e_{k}} \right)}$

where e_(k)=∥P_(model) ^((l))−P_(data) ^((l))∥_(k) is the Euclideandistance between the model tip prediction and the data in experiment k.To implement this minimization, the Nelder-Meade simplex algorithm wasused.

To ensure fair comparison of the coupled model and the point momentmodel, the calibration procedure was performed separately for eachmodel. Results are shown in Table III.

TABLE II EXPERIMENTAL TENSIONS AND TIP LOADS Experiments with Tendons1-4 (Straight) Tension (N) 0 0.98 1.96 2.94 2.94  2.94  4.91 Tip Load(N) 0 0   0   0   0.098 0.196 0   Experiments with Tendon 5 (Helical)Tension (N) 0.98 1.96 2.94 4.91 4.91  4.91  6.87 Tip Load (N) 0   0  0   0   0.098 0.196 0   Experiments with Tendon 6 (Polynomial) Tension(N) 1.50 2.46 3.66 4.91 4.91  Tip Load (N) 0   0   0   0   0.0196Note that the similarity in calibrated Euler angles and their lowdeviation from nominal provides confidence that the correct base framewas obtained for both models. It is also important to note that themodels contain the same number of parameters, no a fair comparison canbe made. As expected, the calibrated values for Young's modulus arehigher than the nominal value of 210 GPa for spring steel, due to theincreased stiffness provided by the disks and glue. Poisson's ratio washeld constant at v=0.3125 during calibration so that the shear moduluswas correctly scaled relative to Young's modulus.

D. Straight Tendon Results and Model Comparison

Table I details the location of the tendon routing paths used in theexperiments in terms of x_(i)(s) and y_(i)(s) as defined in (8).Twenty-five (25) experiments were performed (detailed in Table II) withstraight tendon paths in order to compare the accuracy of the newcoupled model with that of the point moment model. The tip errorstatistics for both models with calibrated parameters is detailed inTable IV.

TABLE IV MODEL TIP ERRORS FOR STRAIGHT TENDON EXPERIMENTS 13 Cases withIn-Plane Loads Tip Error Statistics (mm) mean std. dev. min max PointMoment Model 3.5 1.4 1.2 5.6 Coupled Model 3.1 1.3 0.3 5.3 12 Cases withOut-of-Plane Loads Tip Error Statistics (mm) mean std. dev. min maxPoint Moment Model 9.8 5.5 1.7 16.2 Coupled Model 4.1 2.1 0.6 7.9The results for in-plane loading are accurate for both models, as shownin FIG. 9. FIG. 9 is an x-y-z plot of the actual and simulated result ofin-plane loading of an exemplary continuum robot using a straighttendon. Shown in FIG. 9 are the 13 experimental cases with in-planeexternal loads. The tendons on the top and bottom of the robot (tendons1 and 3) were tensioned and vertical tip loads were applied in four ofthe cases. Distributed gravitational loading is present in every case.As detailed in Table IV, both the coupled model and the point momentmodel are accurate and nearly identical for in-plane loads. in contrast,for out-of-plane loads, the coupled model provides more accuratepredictions, as shown in FIG. 10. FIG. 10 is an x-y-z plot of the actualand simulated result of out-of-plane loading of an exemplary continuumrobot using a straight tendon. Pictured in FIG. 10 are the twelveexperimental cases with out-of-plane external loads. The tendons on theleft and right of the robot (tendons 2 and 4) were tensioned. (a)Distributed loading (robot self-weight) applied, (b) additional tiploads applied. As detailed in Table IV, the data agrees with the coupledmodel prediction, but the point moment model becomes inaccurate as theout-of-plane load increases, and as the curvature increases.

With calibrated parameters, the mean tip error over all 25 straighttendon experiments was 3.6 mm for the coupled model. This corresponds to1.5% of the total arc length of the robot. Note that experimental datapoints lie close to the model prediction along the entire robot length,and the error increases gradually along the robot length, no that tiperror normalized by the robot length is a reasonable metric for theaccuracy of the model.

E. A High-Tension, Large-Load, Straight Tendon Experiment

Also performed was one additional straight tendon experiment to see howthe two approaches compare for a case of large tension and largeout-of-plane load, similar to the case which is simulated in FIG. 1.Tendon 4 was tensioned to 6.38 N and a downward tip force of 0.196 N wasapplied. The resulting data and model predictions are shown in FIG. 11.FIG. 11 is an x-y-z plot of the actual and simulated result ofout-of-plane loading for an exemplary continuum robot using a straighttendon and high tension. As illustrated in FIG. 11, the two modelsproduce very different results. FIG. 11 shows that the coupled modelprediction lies much closer to the data. Here, the tip error of thepoint moment model is 57 mm (23.5% of robot length), while the coupledmodel tip error is 12.8 mm (5.3% of robot length).

F. Experiments with Helical Tendon Routing

To explore more complex tendon routing, helical routing paths were alsoevaluated. As given in Table I, the helical routing path winds throughone complete revolution as it traverses the robot from base to tip. Thetensions and tip loads for these experiments are detailed in Table II.Using the parameters calibrated from the previous straight tendondataset, the resulting data and model predictions are plotted in FIGS.12A and 12B. FIG. 12A is an x-y-z plot of the actual and simulatedresults of operation of an exemplary continuum robot using a helicaltendon without a load. FIG. 12B is an x-y-z plot of the actual andsimulated results of operation of an exemplary continuum robot using ahelical tendon with a tip load. As seen from Table V, the model agreeswith the data with a mean tip error of 5.5 mm. The small increase inerror over the straight tendon cases may be due to increased frictionalforces since the tension for the helical cases was higher.

TABLE V COUPLED MODEL TIP ERRORS FOR NON-STRAIGHT TENDON EXPERIMENTSmean std. dev. min max Tendon 5 (Helical) 5.5 2.7 1.9 10.0 Tendon 6(Polynomial) 4.6 1.9 2.7 7.2

G. Experiments with Polynomial Tendon Routing

In order to further illustrate the model's generality, an additionalexperiment with a general curved tendon routing choice was performed. Inparticular, the routing path variables were parameterized by twotrigonometric functions whose arguments are defined by a polynomialfunction of degree 4 as follows:

x ₆(s)=8 cos(5887s ⁴−2849s ³+320s ²+6s)

y ₆(s)=8 sin(5887s ⁴−2849s ³+320s ²+6s),  (28)

where s is in meters and x₆ and y₆ are in millimeters. This routing pathstarts at the top of the robot, wraps around to the right side for mostof the length, and then returns to the top at the end of the robot. Thetensions and loads are given in Table II, and the results are detailedin Table V and illustrated in FIG. 13. FIG. 13 is an x-y-z plot of theactual and simulated results of operation of an exemplary continuumrobot using a polynomial tendon with tip loads according to Table II.The coupled model's predictions agree with the data, with a mean tiperror of 4.6 mm. This set of experiments confirms the coupled model'sability to handle an arbitrary tendon routing choices.

H. Sources of Error

The largest source of measurement is likely the procedure of manuallyplacing the tip of the on the robot during data capture. It is estimatedthat this uncertainty is at most 2 mm. In general, the largest modelerrors occurred when the tendons were under the greatest tension. Thisagrees with the intuition that effects of static friction should becomemore significant as the tension and curvature increase, However, the lowoverall errors suggest that neglecting static friction is justifiablefor this prototype.

VI. System Configuration

Accordingly, in the view of the foregoing, the equations above can beintegrated into a continuum robot system, as shown in FIG. 14. FIG. 14is a schematic illustration of a continuum robot system 1400 inaccordance with the various embodiments. The system 1400 can include acontinuum robot 1402 similar to that illustrated in FIG. 8. That is, thecontinuum robot 1402 can include an elastic member or backbone 1404having one end or portion coupled to a support or base (not shown), anumber of guide portions 1406, and at least one tendon 1408 extendingthrough the guide portions and defining a tendon path, as describedabove. The system 1402 can further include an actuator/sensor 1410 forapplying a force or tension and for ascertaining a current tension onthe tendon 1408 to the tendon 1408. The system 1400 can also include acontrol system 1412 for operating the system 1400, which can include acomputing device.

The system can have at least two modes of operation. In a first mode ofoperation, the actuator/sensor 1410 can generate signals indicative of acurrent tension on the tendon 1408. This signal can be recited by thecontrol system 1412. The control system 1412 can then use the equationsdescribed above, particularly the governing equations at (17), toestimate a current or resulting shape of the member 1404. In particular,the governing equations at (17) can be solved to extract the shape ofthe member 1404. Additional sensors 1414, such as video sensors, canalso be coupled to the control system 1412 to allow verification of thisestimated shape. In a second mode of operation, the control system 1412can also use the equations described above, particularly the governingequations at (17), to determine an amount of tension required for themember 1404 to achieve a desired shape. Thereafter, the control system1412 can cause the actuator/sensor 1410 to adjust the tension on thetendon 1408. The additional sensors 141 can then be used to verify thatthe target shape has been achieved.

Using these two modes of operation, it is then possible to control therobot 1402 to perform various types of tasks, as the equations aboveallow one to detect and adjust the configuration of the robot 1402 inreal-time based on measurement and adjustment of the tension of thetendon 1408. That is, a robot with increased dexterity can be provided.Such a robot can be useful for various applications. In particular, suchrobots would be useful for carry out procedures in confined spaces, asthe increased dexterity would allow the user to maneuver the tip aroundobstructions in such spaces. For example, such robots could be used toreduce the invasiveness of some existing surgical procedures whichcurrently cannot be performed using conventional robotic tools. Suchprocedures in transnasal skull base surgery, lung interventions,cochlear implantation procedures, to name a few. However, the variousembodiments are not limited in this regard and the various methods andsystems described herein can be used for any other procedure in whichincreased dexterity of the robot is desired or required.

Turning now to FIG. 15, this is shown an exemplary system 1500 includesa general-purpose computing device 1500 for performing one or more ofthe various methods and processes described above. System 1500 includesa processing unit (CPU or processor) 1520 and a system bus 1510 thatcouples various system components including the system memory 1530 suchas read only memory (ROM) 1540 and random access memory (RAM) 1550 tothe processor 1520. The system 1500 can include a cache 1522 of highspeed memory connected directly with, in close proximity to, orintegrated as part of the processor 1520. The system 1500 copies datafrom the memory 1530 and/or the storage device 1560 to the cache 1522for quick access by the processor 1520. In this way, the cache 1522provides a performance boost that avoids processor 1520 delays whilewaiting for data. These and other modules can be configured to controlthe processor 1520 to perform various actions. Other system memory 1530may be available for use as well. The memory 1530 can include multipledifferent types of memory with different performance characteristics. Itcan be appreciated that the disclosure may operate on a computing device1500 with more than one processor 1520 or on a group or cluster ofcomputing devices networked together to provide greater processingcapability. The processor 1520 can include any general purpose processorand a hardware module or software module, such as module 1 1562, module2 1564, and module 3 1566 stored in storage device 1560, configured tocontrol the processor 1520 as welt as a special-purpose processor wheresoftware instructions are incorporated into the actual processor design.The processor 1520 may essentially be a completely self-containedcomputing system, containing multiple cores or processors, a bus, memorycontroller, cache, etc. A multi-core processor may be symmetric orasymmetric.

The system bus 1510 may be any of several types of bus structuresincluding a memory bus or memory controller, a peripheral bus, and alocal bus using any of a variety of bus architectures. A basicinput/output (BIOS) stored in ROM 1540 or the like, may provide thebasic routine that helps to transfer information between elements withinthe computing device 1500, such as during start-up. The computing device1500 further includes storage devices 1560 such as a hard disk drive, amagnetic disk drive, an optical disk drive, tape drive or the like. Thestorage device 1560 can include software modules 1562, 1564, 1566 forcontrolling the processor 1520. Other hardware or software modules arecontemplated. The storage device 1560 is connected to the system bus1510 by a drive interface. The drives and the associated computerreadable storage media provide nonvolatile storage of computer readableinstructions, data structures, program modules and other data for thecomputing device 1500. In one aspect, a hardware module that performs aparticular function includes the software component stored in a tangibleand/or intangible computer-readable medium in connection with thenecessary hardware components, such as the processor 1520, bus 1510,display 1570, and so forth, to carry out the function. The basiccomponents are known to those of skill in the art and appropriatevariations are contemplated depending on the type of device, such aswhether the device 1500 is a small, handheld computing device, a desktopcomputer, or a computer server.

Although the exemplary embodiment described herein employs the hard disk1560, it should be appreciated by those skilled in the art that othertypes of computer readable media which can store data that areaccessible by a computer, such as magnetic cassettes, flash memorycards, digital versatile disks, cartridges, random access memories(RA.Ms) 1550, read only memory (ROM) 1540, a cable or wireless signalcontaining a bit stream and the like, may also be used in the exemplaryoperating environment. Tangible, non-transitory computer-readablestorage media expressly exclude media such as energy, carrier signals,electromagnetic waves, and signals per se.

To enable user interaction with the computing device 1500, an inputdevice 1590 represents any number of input mechanisms, such as amicrophone for speech, a touch-sensitive screen for gesture or graphicalinput, keyboard, mouse, motion input, speech and so forth. An outputdevice 1570 can also be one or more of a number of output mechanismsknown to those of skill in the art. In some instances, multimodalsystems enable a user to provide multiple types of input to communicatewith the computing device 1500. The communications interface 1580generally governs and manages the user input and system output. There isno restriction on operating on any particular hardware arrangement andtherefore the basic features here may easily be substituted for improvedhardware or firmware arrangements as they are developed.

For clarity of explanation, the illustrative system embodiment ispresented as including individual functional blocks including functionalblocks labeled as a “processor” or processor 1520. The functions theseblocks represent may be provided through the use of either shared ordedicated hardware, including, but not limited to, hardware capable ofexecuting software and hardware, such as a processor 1520, that ispurpose-built to operate as an equivalent to software executing on ageneral purpose processor. For example the functions of one or moreprocessors presented in FIG. 15 may be provided by a single sharedprocessor or multiple processors. (Use of the term “processor” shouldnot be construed to refer exclusively to hardware capable of executingsoftware.) Illustrative embodiments may include microprocessor andlordigital signal processor (DSP) hardware, read-only memory (ROM) 1540 forstoring software performing the operations discussed below, and randomaccess memory (RAM) 1550 for storing results. Very large scaleintegration (VLSI) hardware embodiments, as well as custom VLSIcircuitry in combination with a general purpose DSP circuit, may also beprovided.

The logical operations of the various embodiments are implemented as:(1) a sequence of computer implemented steps, operations, or proceduresrunning on a programmable circuit within a general use computer, (2) asequence of computer implemented steps, operations, or proceduresrunning on a specific-use programmable circuit; and/or (3)interconnected machine modules or program engines within theprogrammable circuits. The system 1500 shown in FIG. 15 can practice allor part of the recited methods, can be a part of the recited systems,and/or can operate according to instructions in the recited tangiblecomputer-readable storage media. Such logical operations can beimplemented as modules configured to control the processor 1520 toperform particular functions according to the programming of the module.For example, FIG. 15 illustrates three modules Mod1 1562, Mod2 1564 andMod3 1566 which are modules configured to control the processor 1520.These modules may be stored on the storage device 1560 and loaded intoRAM 1550 or memory 1530 at runtime or may be stored as would be known inthe art in other computer-readable memory locations.

VII. Derivation Appendix

A. Nomenclature

-   *: Denotes a variable in the reference state.-   {dot over ( )}: Denotes a derivative with respect to s.-   ̂: Converts    ³ to so(3) and    ⁶ to se(3):

${\hat{u} = \begin{bmatrix}0 & {- u_{z}} & u_{y} \\u_{z} & 0 & {- u_{x}} \\{- u_{y}} & u_{x} & 0\end{bmatrix}},{\begin{bmatrix}v \\u\end{bmatrix}^{\hat{}} = \begin{bmatrix}0 & {- u_{z}} & u_{y} & v_{x} \\u_{z} & 0 & {- u_{x}} & v_{y} \\{- u_{y}} & u_{x} & 0 & v_{z} \\0 & 0 & 0 & 0\end{bmatrix}}$

-   : Inverse of the ̂ operation.    =u.-   s: ∈    —Reference length parameter.-   p(s): ∈    ³—Position of the robot backbone centroid in global frame    coordinates.-   R(s): ∈SO(3)—Orientation of the robot backbone material with iespect    to the global frame.-   g(s): ∈SE(3)—Homogeneous transformation containing R(s) and p(s).    (The “body frame”)-   r_(i)(s) ∈    ³: Position of the i^(th) tendon with respect to the body frame.    r_(i)(s)=[x_(i)(s) y_(i)(s) 0]^(T)-   p_(i)(s): ∈    ³—Position of the tendon in global frame coordinates.    p_(i)(s)=Rr_(i)(s)+p(s).-   u(s): ∈    ³—Angular rate of change of g with respect to s in body-frame    coordinates. u=-   v(s): ∈    ³—Linear rate of change of q with respect to s expressed in    body-frame coordinates. v=R^(T){dot over (p)}-   n(s): ∈    ³—Internal force in the backbone expressed in global frame    coordinates.-   m(s): ∈    ³—Internal moment in the backbone expressed in global frame    coordinates.-   f_(e)(s): ∈    ³—External force per unit s on the backbone expressed in global    frame coordinates.-   l_(e)(s): ∈    ³—External moment per unit s on the backbone expressed in global    frame coordinates.-   f_(t)(s): ∈    ³—Sum of all forces per unit s applied to the backbone by the    tendons, expressed in global frame coordinates.-   l_(t)(s): ∈    ³—Sum of all moments per unit s applied to the backbone by the    tendons, expressed in global frame coordinates.-   f_(i)(s): ∈    ³—Force per unit s applied to the i^(th) tendon by its surroundings.-   n_(i)(s): ∈    ³—Internal force in the i^(th) tendon.-   τ_(i): ∈    —Tension in the i^(th) tendon. It is constant along s under the    frictionless assumption.

B. Derivation of f_(i)(s)

Beginning with (11),

${n_{i} = {\tau_{i}\frac{{\overset{.}{p}}_{i}}{{\overset{.}{p}}_{i}}}},$

one can re-arrange and differentiate to obtain

${{\overset{.}{p}}_{i} = {\frac{1}{\tau_{i}}{{\overset{.}{p}}_{i}}n_{i}}},{{\hat{p}}_{i} = {\frac{1}{\tau_{i}}{\left( {{\frac{}{s}\left( {{\overset{.}{p}}_{i}} \right)n_{i}} + {{{\overset{.}{p}}_{i}}{\overset{.}{n}}_{i}}} \right).}}}$

Noting that n_(i)×n_(i)=0, one can take a cross product of the tworesults above to find,

${{\overset{¨}{p}}_{i} \times {\overset{.}{p}}_{i}} = {\frac{{{\overset{.}{p}}_{i}}^{2}}{\tau_{i}^{2}}\left( {{\overset{.}{n}}_{i} \times n_{i}} \right)}$

and so

${{\overset{.}{p}}_{i} \times \left( {{\overset{¨}{p}}_{i} \times {\overset{.}{p}}_{i}} \right)} = {\frac{{{\overset{.}{p}}_{i}}^{3}}{\tau_{i}^{3}}{\left( {n_{i} \times \left( {{\overset{.}{n}}_{i} \times n_{i}} \right)} \right).}}$

Applying the vector triple product identity, a×(b×c)=b(a·c)−c(a·b), onecan expand the right-hand side of this equation. Since τ_(i) (themagnitude of n_(i)) is constant with respect to s, then n_(i)·{dot over(n)}_(i)=0 and this results in

$f_{i} = {{- {\overset{.}{n}}_{i}} = {{- \tau_{i\;}}{\frac{{\overset{.}{p}}_{i} \times \left( {{\overset{¨}{p}}_{i} \times {\overset{.}{p}}_{i}} \right)}{{{\overset{.}{p}}_{i}}^{3}}.}}}$

Using the fact that a×b=−b×a, and writing the cross products inskew-symmetric matrix notation in (a×b={dot over (a)}b), one arrives at(12)

$f_{i} = {{\tau_{i}\frac{{\overset{.}{p}}_{i} \times \left( {{\overset{.}{p}}_{i} \times {\overset{¨}{p}}_{i}} \right)}{{{\overset{.}{p}}_{i}}^{3}}} = {\tau_{i\;}\frac{{\overset{\hat{.}}{p}}_{i}^{2}}{{{\overset{.}{p}}_{i}}^{3}}{{\overset{¨}{p}}_{i}.}}}$

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not limitation. Numerous changes to the disclosedembodiments can be made in accordance with the disclosure herein withoutdeparting from the spirit or scope of the invention. Thus, the breadthand scope of the present invention should not be limited by any of theabove described embodiments. Rather, the scope of the invention shouldbe defined in accordance with the following claims and theirequivalents.

Although the invention has been illustrated and described with respectto one or more implementations, equivalent alterations and modificationswill occur to others skilled in the art upon the reading andunderstanding of this specification and the annexed drawings. Inaddition, while a particular feature of the invention may have beendisclosed with respect to only one of several implementations, suchfeature may be combined with one or more other features of the otherimplementations as may be desired and advantageous for any given orparticular application.

For example, in some embodiments, the method described above can also beused to determine the load or external forces and moments being appliedto the elastic member. In such embodiments, the 3D shape of the elasticmember can be determined via some kind of sensing method (i.e., withcameras, or optical fibers, or magnetic tracking coils, or ultrasound,or fluoroscopy, etc.). Thereafter, using a known tension on the tendonand routing path for the tendon, the iterative model equations describedabove can be iteratively solved to determine the external forces andmoments (f_(e) and l_(e)) which result in the model-predicted shape thatis close to the actual sensed shape. The resulting loads based on themodel can then be used as an estimate of the loads acting on the elasticmember. Accordingly, these loads can be used to provide usefulinformation to one who is operating the continuum robot. Alternatively,a similar method can be used to compute the required tendon tensionnecessary to achieve forces and moments for the continuum robot to exerton its surroundings. In such embodiments, the 3D shape of the elasticmember is also determined via some kind of sensing method. Thereafter,the external loads are estimated using the above-mentioned procedure.Finally, the adjustment in tension needed to achieve a desired load orshape can be determined iteratively using the system of equations.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. Furthermore, to the extent that the terms “including”,“includes”, “having”, “has”, “with”, or variants thereof are used ineither the detailed description and/or the claims, such terms areintended to be inclusive in a manner similar to the term “comprising.”

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

What is claimed is:
 1. A continuum robot, comprising: an elastic member; a plurality of guide portions disposed along the length of the elastic member; and at least one tendon extending through the plurality of guide portions, wherein said at least one tendon is arranged to extend through the plurality of guide portions to define a tendon path, wherein the at least one tendon is configured to apply a deformation force to the elastic member via the plurality of guide portions, and wherein the tendon path and an longitudinal axis of the elastic member are not parallel.
 2. The continuum robot of claim 1, wherein the tendon path is substantially helical.
 3. The continuum robot of claim 1, wherein the tendon path is defined by a non-linear function.
 4. The continuum robot of claim 1, further comprising a tendon actuator system, the tendon actuator system comprising: an actuator for applying a tension to the at least one tendon; and a processing element for computing a resulting shape of the elastic member based at least on the applied tension.
 5. The continuum robot of claim 4, wherein the processing element is configured for computing the resulting shape by solving a system of equations defined by: $\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix} \overset{.}{v} \\ \overset{.}{u} \end{bmatrix}}} = {\begin{bmatrix} {D + A} & G \\ B & {C + H} \end{bmatrix}^{- 1}\begin{bmatrix} d \\ c \end{bmatrix}}}$ where u is the deformed curvature vector consisting of the angular rates of change of the attached rotation matrix R with respect to arc length s, v is a vector comprising linear rates of change of the attached frame with respect to arc length s, C and D are stiffness matrices for the elastic member, and matrices A, B, G, H, are functions of the tension applied to the at least one tendon, the tendon path, and its derivatives, d is vector based on the external force on the elastic member, and c is vector based on an external moment on the elastic member.
 6. The continuum robot of claim 5, further comprising at least one sensor for detecting an actual shape of the elastic member after the tension is applied to the at least one tendon, and wherein the processing element is further configured to determining a load on the elastic member based on the system of equations and a difference between the actual shape and the resulting shape.
 7. The continuum robot of claim 4, further comprising a tendon actuator system, the tendon actuator system comprising: a processing element for computing a tensiot o apply to the at least one tendon to achieve a target shape for the elastic member; and an actuator for applying the tension to the at least one tendon.
 8. The continuum robot of claim 7, wherein the processing element is configured for computing the tension for the at least one tendon by evaluating the system of equations given by: $\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix} \overset{.}{v} \\ \overset{.}{u} \end{bmatrix}}} = {\begin{bmatrix} {D + A} & G \\ B & {C + H} \end{bmatrix}^{- 1}\begin{bmatrix} d \\ c \end{bmatrix}}}$ where u is the deformed curvature vec or consisting of the angular rates of change of the attached rotation matrix R with respect to arc length s, v is a vector comprising linear rates of change of the attached frame with respect to arc length s, C and D are stiffness matrices for the elastic member, and matrices A, B, G, H, are functions of the tension applied to the at least one tendon, the tendon path, and its derivatives, d is vector based on the external force on the elastic member, and c is vector based on an external moment on the elastic member.
 9. The continuum robot of claim 8, wherein the processing element is configured for iteratively evaluating the system of equations to compute to the tension.
 10. The continuum robot of claim 8, further comprising at least one sensor for detecting an actual shape of the elastic member after the tension is applied to the at least one tendon, and wherein the processing element is further configured to determine a load on the elastic member based on the system of equations and a difference between the actual shape and the target shape.
 11. A method for managing a continuum robot comprising an elastic member, a plurality of guide portions disposed along the length of the elastic member, and at least one tendon extending through the plurality of guide portions, wherein said at least one tendon is arranged to extend through the plurality of guide portions to define a tendon path, and wherein the at least one tendon is configured to apply a deformation force to the elastic member via the plurality of guide portions, the method comprising: applying a tension to the at least one tendon; and computing the resulting shape of the elastic member resulting from said tension by solving the system of equations given by: $\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix} \overset{.}{v} \\ \overset{.}{u} \end{bmatrix}}} = {\begin{bmatrix} {D + A} & G \\ B & {C + H} \end{bmatrix}^{- 1}\begin{bmatrix} d \\ c \end{bmatrix}}}$ where u is the deformed curvature vector consisting of the angular rates of change of the attached rotation matrix R with respect to arc length s, v is a vector comprising linear rates of change of the attached frame with respect to arc length s, C and D are stiffness matrices for the elastic member, and matrices A, B, G, H, are functions of the tension applied to the at least one tendon, the tendon path, and its derivatives, d is vector based on the external force on the elastic member, and c is vector based on an external moment on the elastic member.
 12. The method of claim 11, further comprising detecting an actual shape of the elastic member after the tension is applied to the at least one tendon, and wherein the processing element is further configured for determining a load on the elastic member based on the system of equations and a difference between the actual shape and the resulting shape.
 13. The method of claim 11, further comprising selecting the tendon path and a longitudinal axis of the elastic member to be non-parallel.
 14. The method of claim 12, further comprising selecting the tendon path to be helical with respect to the longitudinal axis of the elastic member.
 15. The method of claim 12, further comprising selecting the tendon path based on non-linear function.
 16. A computer-readable medium having computer-readable code stored thereon for causing a computer to perform the method recited in any one of claims 11-15.
 17. A method for managing a continuum robot comprising an elastic member, a plurality of guide portions disposed along the length of the elastic member, and at least one tendon extending through the plurality of guide portions, wherein said at least one tendon is arranged to extend through the plurality of guide portions to define a tendon path, and wherein the at least one tendon is configured to apply a deformation force to the elastic member via the plurality of guide portions, the method comprising: determining a target shape for the elastic member; computing a tension for the at least one tendon to provide the target shape by evaluating the system of equations given by: $\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix} \overset{.}{v} \\ \overset{.}{u} \end{bmatrix}}} = {\begin{bmatrix} {D + A} & G \\ B & {C + H} \end{bmatrix}^{- 1}\begin{bmatrix} d \\ c \end{bmatrix}}}$ where u is the deformed curvature vector consisting of the angular rates of change of the attached rotation matrix R with respect to arc length s, v is a vector comprising linear rates of change of the attached frame with respect to arc length s, C and D are stiffness matrices for the elastic member, and matrices A, B, G, H, are functions of the tension applied to the at least one tendon, the tendon path, and its derivatives, d is vector based on the external force on the elastic member, and c is vector based on an external moment on the elastic member.
 18. The method of claim 17, further detecting an actual shape of the elastic member after the tension is applied to the at least one tendon, and wherein the processing element is further configured for determining a load on the elastic member based on the system of equations and a difference between the actual shape and the target shape.
 19. The method of claim 17, ffirther comprising selecting the tendon path and a longitudinal axis of the elastic member to be non-parallel.
 20. The method of claim 19, further comprising selecting the tendon path to be helical with respect to the longitudinal axis of the elastic member.
 21. The method of claim 19, further comprising selecting the tendon path based on a non-linear function.
 22. A computer-readable medium having computer-readable code stored thereon for causing a computer to perform the method recited in any one of claims 17-22.
 23. A continuum robot, comprising: an elastic member; a plurality of guide portions disposed along the length of the elastic member; and at least one tendon extending through the plurality of guide portions and arranged to extend through the plurality of guide portions to define a tendon path, wherein the at least one tendon is configured to apply a deformation force to the elastic member via the plurality of guide portions; an actuator for applying a tension to the at least one tendon; and a processing element for using a system of equations for controlling a shape of the elastic member and the tension, wherein the system of equations comprises: $\overset{.}{p} = {Rv}$ $\overset{.}{R} = {{R{\hat{u}\begin{bmatrix} \overset{.}{v} \\ \overset{.}{u} \end{bmatrix}}} = {\begin{bmatrix} {D + A} & G \\ B & {C + H} \end{bmatrix}^{- 1}\begin{bmatrix} d \\ c \end{bmatrix}}}$ where u is the deformed curvature vector consisting of the angular rates of change of the attached rotation matrix R with respect to arc length s, v is a vector comprising linear rates of change of the attached frame with respect to arc length s, C and D are stiffness matrices for the elastic member, and matrices A, B, G, H, are functions of the tension applied to the at least one tendon, the tendon path, and its derivatives, d is vector based on the external force on the elastic member, and c is vector based on an external moment on the elastic member.
 24. The continuum robot of claim 5, further comprising at least one sensor for detecting an actual shape of the elastic member, and wherein the processing element is further configured to determining a load on the elastic member based on the system of equations, the actual shape, and the applied tension.
 25. The continuum robot of claim 23, wherein the tendon path and a longitudinal axis of the elastic member are not parallel. 